The solution of the differential equation `ydx-xdy+xy^(2)dx=0,` is

Options

A. `(x)/(y)+x^(2)=lambda`
B. `(x)/(y)-x^(2)/(2)=lambda`
C. `(x)/(2y^(2))+x^(2)/(4)=lambda`
D. None of these

Answer

To solve the differential equation \( y \, dx - x \, dy + xy^2 \, dx = 0 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ y \, dx - x \, dy + xy^2 \, dx = 0 \]
We can rearrange this to isolate terms:
\[ xy^2 \, dx = x \, dy - y \, dx \]

### Step 2: Dividing by \( y^2 \)
Next, we divide the entire equation by \( y^2 \):
\[ \frac{xy^2 \, dx}{y^2} = \frac{x \, dy}{y^2} - \frac{y \, dx}{y^2} \]
This simplifies to:
\[ x \, dx = \frac{x \, dy}{y^2} - \frac{y \, dx}{y^2} \]

### Step 3: Recognizing the Derivative
We can rewrite the left-hand side as:
\[ x \, dx = \frac{dy}{y^2} - \frac{y \, dx}{y^2} \]
Now, we can express the left-hand side as a derivative:
\[ d\left(\frac{x}{y}\right) = -x \, dx \]

### Step 4: Integrating Both Sides
Now we integrate both sides:
\[ \int d\left(\frac{x}{y}\right) = \int -x \, dx \]
This gives us:
\[ \frac{x}{y} = -\frac{x^2}{2} + C \]
where \( C \) is the integration constant.

### Step 5: Rearranging the Solution
Rearranging this gives us:
\[ \frac{x}{y} + \frac{x^2}{2} = C \]
or equivalently:
\[ \frac{x}{y} + \frac{x^2}{2} = C \]

### Final Solution
Thus, the solution of the differential equation is:
\[ \frac{x}{y} + \frac{x^2}{2} = C \]